Hard vs soft constraints in the full field reconstruction of incompressible flow kinematics from noisy scattered velocimetry data

High quality flow kinematics reconstruction from noisy and spatially scattered data requires the use of regularization techniques but remains a challenge. We set out to test the effect and practical relevance of additional incompressibility constraints. To this end, we present two methods for reconstructing smooth velocity and velocity gradient fields from such data in an incompressible two-dimensional complex flow. One is based on a generalized Tikhonov regularization combined with a finite element approximation and uses a stream function formulation, which enforces incompressibility (hard constraint). This approach is compared to that in which incompressibility is asymptotically achieved by adding a divergence penalty term in the regularization expression (soft constraint). The methods are compared on synthetic velocity data, obtained for an incompressible Oldroyd–B fluid in a cross-slot channel with added noise. For such data sets, both methods are seen to lead to essentially identical results. However, for a given grid size, the stream function formulation uses a single regularization parameter and less degrees of freedom to provide the required continuity of the gradient fields. The fidelity of the reconstruction is investigated in terms of the quality of the streamlines and velocity gradient history. Incompressibility constraints turn into significant and valuable improvement for applications as we demonstrate by analyzing the stress and optical signal fields obtained by applying a constitutive equation to the reconstructed flow fields.